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Adaptive Deep Density Approximation for Fractional Fokker–Planck Equations

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Abstract

In this work, we propose adaptive deep learning approaches based on normalizing flows for solving fractional Fokker–Planck equations (FPEs). The solution of a FPE is a probability density function (PDF). Traditional mesh-based methods are ineffective because of a unbounded computation domain, a large number of dimensions and a nonlocal fractional operator. To this end, we represent the solution with an explicit PDF model induced by a flow-based deep generative model, which constructs a transport map from a simple distribution to the target distribution. We consider two methods to approximate the fractional Laplacian. One method is the Monte Carlo approximation. The other method is to construct an auxiliary model with Gaussian radial basis functions (GRBFs) to approximate the solution such that we may take advantage of the fact that the fractional Laplacian of a Gaussian is known analytically. Based on these two different ways for the approximation of the fractional Laplacian, we propose two models to approximate stationary FPEs and one model to approximate time-dependent FPEs. To further improve the accuracy, we refine the training set and the approximate solution alternately. A variety of numerical examples is presented to demonstrate the effectiveness of our adaptive deep density approaches.

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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. Ayi, N., Herda, M., Hivert, H., Tristani, I.: On a structure-preserving numerical method for fractional Fokker–Planck equations. arXiv preprint arXiv:2107.13416 (2021)

  2. Brunton, S.L., Noack, B.R., Koumoutsakos, P.: Machine learning for fluid mechanics. Annu. Rev. Fluid Mech. 52, 477–508 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  3. Burkardt, J., Wu, Y., Zhang, Y.: A unified meshfree pseudospectral method for solving both classical and fractional PDEs. SIAM J. Sci. Comput. 43(2), A1389–A1411 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, J., Du, R., Wu, K.: A Comparison Study of Deep Galerkin Method and Deep Ritz Method for Elliptic Problems with Different Boundary Conditions. Commun. Math. Res. 36, 354–376 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, X., Yang, L., Duan, J., Karniadakis, G.E.: Solving inverse stochastic problems from discrete particle observations using the Fokker-Planck equation and physics-informed neural networks. SIAM J. Sci. Comput. 43(3), B811–B830 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  6. Deng, W.: Finite element method for the space and time fractional Fokker–Planck equation. SIAM J. Numer. Anal. 47(1), 204–226 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dinh, L., Krueger, D., Bengio, Y.: NICE: non-linear independent components estimation. arXiv preprint arXiv:1410.8516 (2014)

  8. Dinh, L., Sohl-Dickstein, J., Bengio, S.: Density estimation using real NVP. arXiv preprint arXiv:1605.08803 (2016)

  9. Ditlevsen, P.D.: Observation of \(\alpha \)-stable noise induced millennial climate changes from an ice-core record. Geophys. Res. Lett. 26(10), 1441–1444 (1999)

    Article  Google Scholar 

  10. Elowitz, M.B., Levine, A.J., Siggia, E.D., Swain, P.S.: Stochastic gene expression in a single cell. Science 297(5584), 1183–1186 (2002)

    Article  Google Scholar 

  11. Feng, X., Zeng, L., Zhou, T.: Solving time dependent Fokker–Planck equations via temporal normalizing flow. Commun. Comput. Phys. 32, 401–423 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gao, T., Duan, J., Li, X.: Fokker–Planck equations for stochastic dynamical systems with symmetric lévy motions. Appl. Math. Comput. 278, 1–20 (2016)

    MathSciNet  MATH  Google Scholar 

  13. Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Proceedings of the 13th International Conference on Artificial Intelligence and Statistics, pp. 249–256. JMLR Workshop and Conference Proceedings (2010)

  14. Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., Bengio, Y.: Generative adversarial nets. In: Advances in Neural Information Processing Systems, vol. 27 (2014)

  15. Guo, L., Wu, H., Yu, X., Zhou, T.: Monte Carlo fPINNs: Deep learning method for forward and inverse problems involving high dimensional fractional partial differential equations. Comput. Methods Appl. Mech. Eng. (2022)

  16. Guo, L., Wu, H., Zhou, T.: Normalizing field flows: solving forward and inverse stochastic differential equations using physics-informed flow models. J. Comput. Phys. 461, 111202 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  17. Han, J., Jentzen, A., Weinan, E.: Solving high-dimensional partial differential equations using deep learning. Proc. Nat. Acad. Sci. 115(34), 8505–8510 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. Iten, R., Metger, T., Wilming, H., Del Rio, L., Renner, R.: Discovering physical concepts with neural networks. Phys. Rev. Lett. 124(1), 010508 (2020)

    Article  Google Scholar 

  19. Kingma, D.P., Ba, J.: Adam: a method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014)

  20. Kingma, D.P., Dhariwal, P.: Glow: generative flow with invertible 1x1 convolutions. arXiv preprint arXiv:1807.03039 (2018)

  21. Kingma, D.P., Welling, M.: Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114 (2013)

  22. Liu, S., Li, W., Zha, H., Zhou, H.: Neural parametric Fokker–Planck equations. SIAM J. Sci. Comput. 60(3), 1385–1449 (2022)

    MathSciNet  MATH  Google Scholar 

  23. Meng, X., Karniadakis, G.E.: A composite neural network that learns from multi-fidelity data: application to function approximation and inverse PDE problems. J. Comput. Phys. 401, 109020 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pang, G., Lu, L., Karniadakis, G.: fPINNs: fractional physics-informed neural networks. SIAM J. Sci. Comput. 41, A2603–A2626 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Papamakarios, G., Nalisnick, E., Rezende, D.J., Mohamed, S., Lakshminarayanan, B.: Normalizing flows for probabilistic modeling and inference. J. Mach. Learn. Res. 22, 1–64 (2021)

    MathSciNet  MATH  Google Scholar 

  26. Qin, T., Chen, Z., Jakeman, J.D., Xiu, D.: Deep learning of parameterized equations with applications to uncertainty quantification. Int. J. Uncertain. Quantif. 11(2) (2021)

  27. Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686–707 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  28. Raissi, M., Yazdani, A., Karniadakis, G.E.: Hidden fluid mechanics: learning velocity and pressure fields from flow visualizations. Science 367(6481), 1026–1030 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rezende, D., Mohamed, S.: Variational inference with normalizing flows. In: International Conference on Machine Learning, pp. 1530–1538. PMLR (2015)

  30. Sheng, H., Yang, C.: PFNN: A penalty-free neural network method for solving a class of second-order boundary-value problems on complex geometries. J. Comput. Phys. 428, 110085 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sheng, H., Yang, C.: PFNN-2: A domain decomposed penalty-free neural network method for solving partial differential equations. arXiv preprint arXiv:2205.00593 (2022)

  32. Shlesinger, M., Zaslavsky, G., Frisch, U.: Lévy Flights and Related Topics in Physics. Lecture Notes in Physics, Springer, Berlin (1995)

    Book  MATH  Google Scholar 

  33. Sirignano, J., Spiliopoulos, K.: DGM: A deep learning algorithm for solving partial differential equations. J. Comput. Phys. 375, 1339–1364 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sukumar, N., Srivastava, A.: Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks. Comput. Methods Appl. Mech. Eng. 389, 114333 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  35. Tang, K., Wan, X., Liao, Q.: Deep density estimation via invertible block-triangular mapping. Theor. Appl. Mech. Lett. 10(3), 143–148 (2020)

    Article  Google Scholar 

  36. Tang, K., Wan, X., Liao, Q.: Adaptive deep density approximation for Fokker–Planck equations. J. Comput. Phys. 457, 111080 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  37. Tang, K., Wan, X., Yang, C.: DAS-PINNs: a deep adaptive sampling method for solving high-dimensional partial differential equations. J. Comput. Phys. 476, 111868 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  38. Terrell, G.R., Scott, D.W.: Variable kernel density estimation. Ann. Stat. 1236–1265 (1992)

  39. Weinan, E., Yu, B.: The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Commun. Math. Stat. 6(1), 1–12 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  40. Xu, Y., Zan, W., Jia, W., Kurths, J.: Path integral solutions of the governing equation of SDEs excited by Lévy white noise. J. Comput. Phys. 394, 41–55 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  41. Yang, L., Meng, X., Karniadakis, G.E.: B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data. J. Comput. Phys. 425, 109913 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  42. Yang, L., Zhang, D., Karniadakis, G.E.: Physics-informed generative adversarial networks for stochastic differential equations. SIAM J. Sci. Comput. 42(1), A292–A317 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  43. Yang, Y., Perdikaris, P.: Adversarial uncertainty quantification in physics-informed neural networks. J. Comput. Phys. 394, 136–152 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zan, W., Xu, Y., Kurths, J., Chechkin, A., Metzler, R.: Stochastic dynamics driven by combined Lévy-Gaussian noise: fractional Fokker–Planck–Kolmogorov equation and solution. J. Phys. A Math. Theor. 53 (2020)

  45. Zang, Y., Bao, G., Ye, X., Zhou, H.: Weak adversarial networks for high-dimensional partial differential equations. J. Comput. Phys. 411, 109409 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zhang, H., Jiang, X., Yang, X.: A time-space spectral method for the time-space fractional Fokker–Planck equation and its inverse problem. Appl. Math. and Comput. 320, 302–318 (2018)

    MathSciNet  MATH  Google Scholar 

  47. Zhang, H., Xu, Y., Li, Y., Kurths, J.: Statistical solution to SDEs with \(\alpha \)-stable Lévy noise via deep neural network. Int. J. Dyn. Control 8(4), 1129–1140 (2020)

    Article  MathSciNet  Google Scholar 

  48. Zhang, L., Han, J., Wang, H., Car, R., Weinan, E.: Deep potential molecular dynamics: a scalable model with the accuracy of quantum mechanics. Phys. Rev. Lett. 120(14), 143001 (2018)

    Article  Google Scholar 

  49. Zhu, Y., Zabaras, N., Koutsourelakis, P.S., Perdikaris, P.: Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data. J. Comput. Phys. 394, 56–81 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported by the NSF of China (No. 12288201), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA25010404), the National Key R &D Program of China (2020YFA0712000), the youth innovation promotion association (CAS), and Henan Academy of Sciences. The second author is supported by NSF Grant DMS-1913163.

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Authors and Affiliations

  1. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences, Beijing, China

    Li Zeng & Tao Zhou

  2. School of Mathematics and Statistics, Fuzhou University, Fuzhou, China

    Li Zeng

  3. Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, 70803, USA

    Xiaoliang Wan

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  1. Li Zeng
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Zeng, L., Wan, X. & Zhou, T. Adaptive Deep Density Approximation for Fractional Fokker–Planck Equations. J Sci Comput 97, 68 (2023). https://doi.org/10.1007/s10915-023-02379-z

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